This set is non-empty, convex and closed with respect to the topology of point-wise convergence of bounded linear real functionals, and it furthermore has \(\underline{P}_{}\)  and \(\smash{\overline{P}_{}}\)  as its lower and upper envelopes [1]}:
\(\underline{P}_{}(f)=\min []{P_{}(f)\colon P_{}\in \mathbf {P}(\underline{P}_{})}\text{ and }\overline{P}_{}(f)=\max []{P_{}(f)\colon P_{}\in \mathbf {P}(\underline{P}_{})}.\) 
